Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 36, pp. 1-18.

Title: Weak solution by the sub-supersolution method for a nonlocal system
involving Lebesgue generalized spaces
       
Authors: Abdolrahman Razani (Imam Khomeini International Univ., Qazvin, Iran)
         Giovany M. Figueiredo (Univ. de Brasilia, Brasilia, Brazil)

Abstract:  
We consider a system of nonlocal elliptic equations
$$
\aligned
&- \mathcal{A}(x, |v|_{L^{r_1(x)}})
 \operatorname{div}(a_1(|\nabla u|^{p_1(x)})|\nabla u|^{p_1(x)-2}\nabla u)\\
&= f_1(x, u,v)  |\nabla v|^{\alpha_1(x)}_{L^{q_1(x)}}+g_1(x, u,v)  
|\nabla v|^{\gamma_1(x)}_{L^{s_1(x)}},\\
& - \mathcal{A}(x, |u|_{L^{r_2(x)}})
 \operatorname{div}(a_2(|\nabla v|^{p_2(x)})|\nabla u|^{p_2(x)-2}\nabla u)\\
&= f_2(x, u,v)  |\nabla u|^{\alpha_2(x)}_{L^{q_2(x)}}+g_2(x, u,v)
  |\nabla u|^{\gamma_2(x)}_{L^{s_2(x)}},
\endaligned
$$
with Dirichlet boundary condition, where $\Omega$ is a bounded domain in
$\mathbb{R}^N$ $(N >1)$ with  $C^2$ boundary. Using sub-supersolution method,
we prove the existence of at least one positive weak solution.
Also, we study a generalized logistic equation and a sublinear system.

Submitted June 18, 2021. Published May 01, 2022
Math Subject Classifications: 35J91, 35J60, 35D30.
Key Words: Nonlocal problem, $p(x)$-Laplacian, sub-supersolution; minimal wave speed.